What is Lévy measure, Lévy triplet, characteristic exponent?

The Lévy measure of a Lévy process is $\nu(A) = \E[\#\{s \le 1 : \Delta X_s \in A\}]$, the expected number of jumps per unit time with size in $A$ ($0 \notin \bar A$); it satisfies $\int\min(1, x^2)\,\nu(dx) < \infty$. The Lévy triplet is $(\sigma^2, \nu, \gamma)$: Gaussian variance, Lévy measure and drift. The characteristic exponent is the function $\psi$ with $\varphi_{X_t}(u) = e^{t\psi(u)}$.